In this section, students begin their exploration of exponential functions. They consider patterns of numbers that increase by a common factor and compare them to patterns that increase by a common difference. This motivates the need for a way to describe these non-linear patterns with exponential expressions.

In this section, students revisit comparisons between linear and exponential change, now with the focus on functions. They see that exponential functions eventually grow more quickly than do linear functions. They also justify that functions of the form have a constant difference of between any 2 inputs that are 1 unit apart, while functions of the form have a constant ratio of between any 2 inputs that are 1 unit apart.

Graph of an increasing exponential function, x y plane, origin O. Horizontal axis, scale 0 to 10, by 5’s. Vertical axis, scale 0 to 60, by 20’s. The function starts at the vertical axis and goes through (3 comma 8), (4 comma 16) and (5 comma 32).

In this final section, students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions.

In this section, students apply their understanding of exponential change to situations involving percent increase and decrease. In particular, students explore the context of compounding interest in depth including application of interest at different frequencies.

To simplify the math, many of the situations are not realistic. An initial amount of money is deposited and left untouched for years or a loan is undertaken with no payments made. Be ready to engage students in conversation about more realistic scenarios.

The first lesson of the section provides some optional review of percent change. If the Check Your Readiness assessment shows students have a good understanding of the concept or if their understanding of exponential functions is good from this unit, the lesson could be safely skipped.

In this section, students write expressions to represent exponential growth. They make use of properties of exponents and repeated multiplication of a growth factor. They are introduced to patterns that decrease in an exponential way, by looking at growth factors between 0 and 1 and by rewriting expressions using the distributive property.

Later in the section, students examine graphs of exponential patterns and patterns to consider the meaning of negative exponents in context. Students must rely on properties of exponents, first developed in grade 8, to create a consistent understanding of negative exponents.

Throughout the section, students continue to make connections between descriptions of situations, tables of values, graphs, and expressions or equations representing exponential change. In particular, they explore the meaning of and in the expression using the different representations.

In this section, students begin writing exponential relationships as functions. They revisit function notation and apply language associated with functions, such as dependent and independent variables. They also consider average rate of change in an exponential context and conclude that these measures are less representative of the function than when using linear functions.

Later, students examine the impact of parameters and on functions of the form and apply that in order to mathematically model situations that change exponentially.

The last lesson in the section is optional and provides additional practice exploring the parameters in functions of the form .

Graph of a function on grid, origin O. Horizontal axis, number of bounces, from 0 to 12, by 1's. Vertical axis, height in centimeters, from 0 to 150, by 25's. Line of given equation, h equals 142 time 2 thirds to the n, is graphed, passing through 0 comma 142, 1 comma 94 and two thirds, 2 comma 63 and 1 ninth. 3 comma 42 and 2 twenty-sevenths, 4 comma 28 and 4 eighty-firsts. Data points of 1 comma 95, 2 comma 61, 3 comma 39, 4 comma 26 also plotted.